四川师范大学2009至2010学年度第一学期常微分方程期末考试试题B四川师范大学2009至2010学年度第一学期常微分方程期末考试试题B.doc

四川师范大学2009至2010学年度第一学期常微分方程期末考试试题B 四川师范大学数软学院数学与应用数学期末考试 2008－2009学年度第一学期期末考试 TEST PAPER OF ORDINARY DIFFERENTIAL EQUATIONS (B) 答卷说明：本试卷共6个大题，满分100分，120分钟完卷。 　 Question number I II III IV V VI total signature scores scores signature ? I．(15 points). Solve the first order implicit differential equation . scores signature II．(15 points) Solve the linear differential equations . scores signature III．(15 points).Solve the differential equation . scores signature IV．(15 points) Solve the linear differential equation with the initial-value . scores signature V．(20 points)．If and , discuss of the and the stability of the trivial solution of the system . scores signature VI．(20 points) Prove that is a general solution of the linear differential equation of first order . 常微分方程试题B参考答案及评分细则 I．Solution . Let then the ordinary differential equation is rewrited , 　　　　(1) Differentiate the above equation with respect to x，then 　　　　.　 　　　 5 points It follows that .　 10 points From (1), we have or The solutions of the ordinary differential equation are or . 　15 points II．Solution . Consider the coefficients matrix. The characteristic equation of A is det（）=0. =. The matrix A has three eigenvalues are . 5 points When ，from , we find . When ，from ，we find . . The fundamental matrix solution of is . 10 points Then, . The particular solution of the equation is =. The solution of linear ordinary differential equations is . 15points III．Solution . Let P=2xy,Q=-3, then . 5 points The equation has integrating factor . Multiply the above equation by , then . 10 points Integrate the above equality , it follows that . 15 points IV．Solution. The characteristic equation of the linear differential equation is T